Force on charged particles in a magnetic field

Key ideas

A current is just moving charge, so a single moving charge in a magnetic field feels a force too.

The force relationship

$F = qvB$

• $F$ — force on the charge (N)
• $q$ — charge (C)
• $v$ — speed of the charge (m s⁻¹)
• $B$ — magnetic field strength (T)

This is the maximum force, when the charge moves at right angles to the field.

Circular motion in a field

• The force is always perpendicular to the velocity, so it never speeds the charge up — it only changes its direction.
• A charge moving at right angles to a uniform field therefore travels in a circle.
• The magnetic force provides the centripetal force: $qvB = \dfrac{mv^2}{r}$, which rearranges to a radius $r = \dfrac{mv}{qB}$.
• A charge moving parallel to the field feels no force ($v$ along $B$).

:::tip A magnetic force can never change a charge's speed or kinetic energy — because it always acts at right angles to the motion, it does no work. If a question claims a magnetic field speeds a charge up, that is wrong; it only bends the path. :::

Practice question

A charge of $3.2 \times 10^{-19}$ C moves at $1.0 \times 10^{6}$ m s⁻¹ across a $0.20$ T field. Find the force.

Worked solution: $F = qvB = (3.2 \times 10^{-19})(1.0 \times 10^{6})(0.20) = 6.4 \times 10^{-14}$ N.